Snell's Law & Total Internal Reflection

Q: How does Snell's law follow from Fermat's principle?

Fermat's principle says light takes the path of least (or stationary) time between two points. Writing travel time as a function of the crossing point on the interface and setting its derivative to zero reduces the geometry to n1·sin(theta1) = n2·sin(theta2) — Snell's law falls out of a pure optimisation, with no separate assumption about wave behaviour needed.

Q: What exactly is the critical angle?

Going from a denser medium (n1) into a rarer one (n2 less than n1), sin(theta2) grows faster than sin(theta1). At the critical angle theta_c = arcsin(n2/n1), sin(theta2) reaches exactly 1, meaning the refracted ray would skim along the interface. Beyond that angle no refracted ray exists mathematically, so all light reflects back — total internal reflection.

Q: Is anything happening on the far side of a totally internally reflected surface?

Yes — Maxwell's equations require the field to stay continuous across the boundary, so a non-propagating evanescent wave exists just beyond the surface, decaying exponentially within about one wavelength. If a second medium is brought close enough, this field can couple energy across the gap, an effect called frustrated total internal reflection, used in beam-splitter cubes and fingerprint scanners.

Light bends because it slows down unevenly

Light travels at c = 299,792,458 m/s in vacuum but slows to about 0.75c in water and 0.66c in glass — a slowdown captured by the refractive index n = c/v. When a wavefront crosses a boundary at an angle, one side enters the slower medium before the other and gets "held back," rotating the direction of travel — the same way a marching band bends when one file steps onto mud while the rest is still on pavement. A ray bends toward the normal entering a denser medium, and away from it leaving one.

Deriving Snell's law from least time

Fermat's principle states that light travels between two points along the path of least (or stationary) time. Writing the total travel time as a function of the crossing point on a flat interface and setting its derivative to zero collapses a messy geometric optimisation into one clean relationship between the sines of the incidence and refraction angles.

n₁ sin θ₁ = n₂ sin θ₂                         (Snell's law)
n = c/v      air: 1.0003    water: 1.33    common glass: 1.5–1.9

live demo · a ray bending as it crosses an interface● LIVE

Beyond the critical angle: total internal reflection

Going from a denser medium (n₁) into a rarer one (n₂ < n₁), sin θ₂ grows faster than sin θ₁. At the critical angle θ_c = arcsin(n₂/n₁), sin θ₂ reaches exactly 1 and the refracted ray would skim along the interface itself; beyond that angle there is no mathematical solution for θ₂ at all — every photon reflects back into the denser medium, with 100% efficiency and no absorption. Diamond's tiny critical angle of 24.4° (versus 48.6° for water and 41.1° for crown glass) is why cut diamonds trap light in repeated internal bounces before it escapes toward the viewer, and it's the identical physics that keeps light bouncing down an optical fibre for thousands of kilometres.

sin θ_c = n₂ / n₁
Water (n=1.33) → air: θ_c ≈ 48.6°
Diamond (n=2.42) → air: θ_c ≈ 24.4°

Frequently asked questions

How does Snell's law follow from Fermat's principle?

Fermat's principle says light takes the path of least (or stationary) time between two points. Writing travel time as a function of the crossing point on the interface and setting its derivative to zero reduces the geometry to n1·sin(theta1) = n2·sin(theta2) — Snell's law falls out of a pure optimisation, with no separate assumption about wave behaviour needed.

What exactly is the critical angle?

Going from a denser medium (n1) into a rarer one (n2 less than n1), sin(theta2) grows faster than sin(theta1). At the critical angle theta_c = arcsin(n2/n1), sin(theta2) reaches exactly 1, meaning the refracted ray would skim along the interface. Beyond that angle no refracted ray exists mathematically, so all light reflects back — total internal reflection.

Is anything happening on the far side of a totally internally reflected surface?

Yes — Maxwell's equations require the field to stay continuous across the boundary, so a non-propagating evanescent wave exists just beyond the surface, decaying exponentially within about one wavelength. If a second medium is brought close enough, this field can couple energy across the gap, an effect called frustrated total internal reflection, used in beam-splitter cubes and fingerprint scanners.

Try it live

Everything above runs in your browser — open Snell's Law — Refraction & TIR and set n₁, n₂ and the angle of incidence to watch the refracted ray bend or trigger total internal reflection with its evanescent wave. Nothing is installed, nothing is uploaded.

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